Characteristic Flows on Signed Graphs and Short Circuit Covers

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13–28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which are known to occur in two types – either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph G admitting a nowhere-zero k-flow has a covering with signed circuits of total length at most 2(k − 1)|E(G)|.